I have all these problems on occasion on Windows, too. They're just problems with Big Picture mode.
The first one tho, with the controller wonkiness, is usually caused by leaving Steam Input enabled in a game that 100% natively supports your controller. If the game shows a totally filled in icon (as opposed to no icon or a half filled icon) of your controller, you should disable Steam Input and just let the game handle it. You may also want to disable Steam from translating controller input for the OS itself; sometimes it doesn't automatically change to just controlling the running game and still messes around like you're on the desktop.